Consider an experiment that has been repeated N times, where N is a large number, and produced the individual outcomes ni li= ,, , , , , N . Let the frequency of occurrence of ni be P,,.* If one plots P,, versus n,, the resulting curve
' ~ x c e ~ t i o n : Radiation counting measurements with rn < 20 obey the Poisson distribution.
'1f N = 1000 and ni has occurred 15 times, P,, = 15/1000.
50 MEASUREMENT AND DETECTION OF RADIATION
resembles a Gaussian distribution as shown in Fig. 2.10. The larger the value of N, the more the histogram of Fig. 2.10 coincides with a normal distribution.
Assume that the dashed line of Fig. 2.10 is an acceptable representation of the experimental results. Under these circumstances, how should the result of the measurement be reported and what is its standard error?
The result of the measurement is reported as the arithmetic average defined by
This equation is the same as Eq. 2.31. As N increases, a better estimate of the true value of n is obtained-i.e., the error of the measurement becomes smaller. The true value of n, which is also called the true mean, can only be obtained with an infinite number of measurements. Since it is impossible to perform an infinite number of trials, n is always calculated from Eq. 2.71.
The error of E depends on the way the individual measurements are distributed around 5-ie., it depends on the width of the Gaussian of Fig. 2.10.
As the width becomes smaller, the error gets smaller, and therefore the measurement is better. The standard error of E is defined in terms of the standard deviation of the distribution. Using Eq. 2.34 and setting f(xi) = 1 / N , the standard deviation of the distribution becomes
With a finite number of measurements at our disposal, this equation for u has to be modified in two ways. First, because the true mean m is never known, it is replaced by its best estimate, which is ii (Eq. 2.71). Second, it can be generally
/
\
\
\ Figure 2.10 The distribution of
\ the frequency of occurrence of individual results of a series of
1, \ identical follow a Gaussian measurements distribution. tends to
shown that the best estimate of the standard deviation of N measurements is given by the following equation:
The differences between Eq. 2.72 and Eq. 2.73 are the use of Ti instead of m and the use of N - 1 in the denominator instead of N.+ For a large number of measurements, it does not make any practical difference if one divides by N or N - 1. But it makes a difference for small values of N. Using the extreme value of N = 1, one can show that division by N gives the wrong result. Indeed, dividing by N, one obtains
Zero a means zero error, which is obviously wrong. The error is never zero, certainly not in the case of one measurement. Division by N - 1, on the other hand, gives
which, being indeterminate, is a more realistic value of the error based on a single measurement.
Since the N results are distributed as shown in Fig. 2.10,68.3 percent of the outcomes fall between Z - a and Ti + a (see Eq. 2.62). Therefore, one addi- tional measurement has a 68.3 percent chance of providing a result within Ti + a . For this reason, a is called the standard deviation or the standard error of a single measurement. Is this equal to the standard error of Z? No, and here is why.
According to the definition of the standard error, if a, is the standard error of T i , it ought to have such a value that a new average 7i would have a 68.3 percent chance of falling between Ti - uE and n + uE. To obtain the standard error of Z, consider Eq. 2.71 as a special case of Eq. 2 . 3 6 ~ . The quantity Ti is a linear function of the uncorrelated random variables n , , n , , . . . , n,, each with standard deviation a . Therefore
'The factor N - 1 is equal to the "degrees of freedom" or the number of independent data or equations provided by the results. The N independent outcomes constitute, originally, N indepen- dent data. However, after Ti is calculated, only N - 1 independent data are left for the calculation of u.
52 MEASUREMENT AND DETECTION O F RADIATION
where ai = 1 / N . Using Eq. 2.41, the standard deviation of iz ist
If the series of N measurements is repeated, the new average will probably be different from E, but it has a 68.3 percent chance of having a value between iz - u, and iz + uz. The result of the N measurements is
When a series of measurements is performed, it would be desirable to calculate the result in such a way that the error is a minimum. It can be shown that the average E as defined by Eq. 2.71 minimizes the quantity
which is proportional to the standard error. Finally, Eq. 2.75 shows that the error is reduced if the number of trials increases. However, that reduction is proportional to 1/ fi, which means that the number of measurements should be increased by a factor of 100 to be able to reduce the error by a factor of 10.